What math research feels like (for me)
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- Опубликовано: 12 мар 2021
- People usually think math is a dry uncreative subject. It's really not at all though. Doing math research was surprisingly emotional. There's a joy to doing it that's hard to explain.
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Yet, the sad thing is that the general public often think that maths is just about numbers, computations, and so on, completely missing out how challenging, yet rewarding, mathematical discovery could be.
It's crazy. Someone once asked me if I just add bigger and bigger numbers. People have no idea what maths is
@@LookingGlassUniverse "Since you're majoring in Maths, count how many people came to the university choir rehearsal." Actual thing I was told, without any irony. About 300 people in that choir, too.
Math boi: I like maths
Non math boi: oH sO u LiKe NuMbErS
@@LookingGlassUniverse I tell people that arithmetic is to maths as spelling is to literature.
@@LookingGlassUniverse wow that's kinda dumb. Obviously you're multiplying.
It's an odd thing the majority of people only equate creativity and imagination with the arts. Yet every scientific discovery is the consequence of someone's imagination and thinking differently. I could pursue a career in music or literature and never create anything new. To create or find nothing new would be nigh on impossible with a career in the likes of mathematics or physics.
I think this is in the same vein as the misconception that deeper, particularly scientific understanding takes away from the wonders of the world instead of adding onto them aspects of which you were previously unaware.
Funny enough, I majored in art and can tell you that you can get an art degree without being creative. I happened to be better than most people at rendering and executing artistic ideas (ie, I could draw, paint, and sculpt well enough to impress people). But true creativity is in the ideas themselves, which is why I was a "B" student in my art classes while there were "A" students who did work that didn't necessarily look better than mine but their ideas were more interesting. Also, I began undergrad as a math major and can also tell you that once you get to calculus, math becomes less about standard processes for solving problems and more about finding creative ways to solve problems. That is why in the higher levels of math you really need to wrap your head around theory in order to progress to the next level of math because following basic cookie cutter processes no longer work the way they did prior to calculus, differential equations, or linear algebra.
I was given the chance to be taken under a professional flamenco guitarist's wing to become a professional myself, but opted to pursue mathematics and computer science instead, given how limited our time in this life is. I even stopped playing guitar to concentrate myself fully on math and CS. It has been 8 years, and I have not for a second regretted that choice. The beauty, spirituality and creativity that mathematics has brought to my life is indescribable to someone not versed themselves in mathematics. I'm always reminded of Lockhart's Lament when I see people share the same vision. If we taught music or art the same way we do math, most people would hate music and art. www.maa.org/sites/default/files/pdf/devlin/LockhartsLament.pdf
@@mathephilia completely agree, I also share this essay wherever applicable
I just got a contract with a publisher for my first novel in the same month I got accepted into a Masters of Science doing Physics. I was driven to do both by imagination and a sense of wonder. I think too many people treat the arts and sciences as somehow mutually exclusive. It’s nonsense.
I've often described mathematicians as epiphany chasers. The thrill of discovering something new entirely from their imagination is intoxicating. I'm not certain I could stop even if I tried.
"There's nothing like being obsessed with a mathematical problem and have the pictures invade your mind and pursue you down the street." This statement cannot be improved upon.
This happens me with coding too. I will often be talking with my wife, (or doing anything really) then drift away in my mind obsessed with something I'm trying to solve. It really does invade your mind, constantly.
Einstein said that he was living in Princeton just so that he could take long walks and chat about math with Godel
@Katie Ellen Tu I thought that but when I tried to measure it... it changed....
It's so cool to see someone as charismatic as yourself promoting and discussing real mathematics, not to say that talking about "cool results" from mathematics isn't what it's about, but I've never had anyone tell me the personal story of pursuing mathematics.
For me, my bachelor thesis was a minefield of "I think I have it now" to discover, or most often get it pointed out to me, that there was a mistake in the proof somewhere that made the whole thing fall apart. In the end, I ended up with on new result in the direction I was looking at, though it wasn't the airtight iff I was dreaming of. I haven't looked back at anything even remotely related to the topic I worked on because of how much that 5 month period wore me down, but I feel more invigorated to take another look at it, as the only reason my research stopped then was the deadline coming to chop me up. I think the time pressure also lead me to submit proofs to my superiors faster than I would have done naturally.
I'm so glad you are going to look at the work you did again. I know I didn't feel good at all about a lot of my research because, like you said, I couldn't get the airtight results I actually wanted. But looking back on my work I feel pretty proud to have stuck with it now, and the proofs that seemed "trivial" now look so much cooler.
What do you do now?
@@LookingGlassUniverse I'm still studying for my master's degree. Not long ago I caught myself trying to figure out proof about something that was really not part of the curriculum, and I have to say, your video on doing a PhD is starting to grow some doubts in my mind, I used to think I wouldn't pursue anything past a master but now I'm not so sure anymore.
It's really great that you took time to share your story about what real academia looks like and the entire journey of a result. More often than not, we only come to know about results without knowing the details and hardwork that went along the way. It's really fascinating to hear all those details and insights from someone who is really into the field. Also you have articulated your experience pretty nicely which really sheds light on the unnoticed part of proving a result. Thank you so much for doing this. Would love to hear more about your research experiences.
Thank you so much!
Thank you for this detailed description of the process and the genuine excitement you convey. Your chanel is awesome.
So pleased to see you back. Missed your beautiful presentations and candid lucidity. More of this channel please!
This was so enlightening, thank you Mithuna! The mathematical adventure is certainly amazing
Thank you for this, it was so inspiring! It's what I needed to hear while I'm in the depths of study.
This was so awesome to watch, thank you! I think two years ago when I was going crazy deciding what I wanted to study in graduate school, I asked you for what your experience was in the big picture and in the daily grind - and this is exactly that. :) Just so you know, I am now studying synthetic inorganic chemistry which also heavily utilizes creativity and theorizing about steps toward a goal. Kinda thankful now after your video that I can test those steps experimentally and verify them using spectroscopy techniques ;)
This is wonderful, thank you for sharing and illuminating the path ahead for those of us coming up behind you. I used to believe I was not a "math person", but slowly over time i realized that, difficult as a new topic may be at first, ultimately it is learnable.
Thank you Mithuna so so much! Now research seems to be the most fun thing to do. Thanks for guiding about the thought process to prove an idea, very very helpful.
this is *_insanely_* real, and the feelings can be *_very_* intense. pushing through is the hardest/best part.
It's hard to explain to people that (math) research is so emotional, but it really is.
@@LookingGlassUniverse hi Mithuna, thanks for your content and dedication and for being so authentic. I find you and your videos inspiring, and feel I'm on a similar path to you academically. I'm at a bit of a crossroads in life and (well, regardless of my current status) I'd love more videos like this.
Although my background is empirical research, I can so much relate to your story and experiences. And failing faster is definitely a sign of progress. Thanks for sharing your thoughts in such a clear way!
It's really nice to see that you're getting back with the channel.
Thanks a lot for your content!!
Thank you very much! I hope to do a lot more
@LGS squeeeeeee ☆*:. o(≧▽≦)o .:*☆
It's eye-opening to be getting to know about your math journey! In my mind math researchers are mysterious wizards.
I look forward to your next publication story!
That was absolutely fascinating. Thank you. I think all research is like that. In the likes of biology maybe there's less rigour or certainty but plenty of creativity. Some of it absolutely novel, as in your example or it can be the realisation that previous "separate" bits of creativity and rigour are actually related forming a bigger picture. Anyway thanks again and all the best in your future endeavours.
This is like the best video on RUclips for me right now. Thank you so much for this!
Thank you for this beautiful description of mathematical research, from your perspective! You sure know how to explain things very clearly, and, even if it isn't apparent, your choice of words is very precise and concise. That's funny, as I'm doing my own research at an amateur level, and I can totally relate to your experience. I've experienced the total joy of joining two dots, and also the case where reality resists at the last step and sends you back to the start. And also, as you point out the usefulness of those failed attempts. I must say, you seem so overflown with joy and happiness when talking about your passion that you nearly made my heart explode ! And - if you accept compliments - it's so rare to see such a combination of a beautiful mind and an expressive physical beauty. Cheers, and all the best with your passions!
I once had a friend who, upon hearing that I wanted to study math, said to me in a crestfallen tone, "Why don't you do something creative instead?" I started to rebut her, but realized it would be pointless.
Thanks for sharing your experience at trying so hard but failing in the end. The vast majority of the stories out there are success stories, but I think there are also a lot of things that we can learn (perhaps even more than from successes) from the failures.
"No one will cast us from the paradise which Cantor has created for us"--David Hilbert
thank you for sharing this. the experience of math is so gatekept and mysterious that sharing this helps so much ✨
I did my undersgraduate in maths and I'll never forget how I felt when I solved a question in abstract math or abstract algebra. I could feel my creativity at work (even though I wan't creating something new to the world, just a minor proof) and I was surprised how no one had mentioned this side of mathematics before.
Yes, exactly how I felt in ug, like I'd been lied to about the nature of maths!
I was lucky enough to see the power and beauty of abstraction before college. I will begin my studies in college next year. Guess what I will major in?
@@kaanylmaz9825 Good choice! Best of luck to you!
@@kaanylmaz9825 can you explain me (dummy) what do u mean by this statement "power and beauty of abstractions"
I just understand this definition of abstraction :- a general idea not based on any particular real person, thing or situation
Great vid as always! I would love a video covering every book on that bookshelf behind you, your thoughts, if/how they helped you, and if you would recommend them.
Great video, I used to not like maths because of the way it's taught at school, but I started studying it more in depth recently, from different points of view, and I'm in love with it, it involves so much creativity!
I really appreciate you sharing your stories of both successes and failures, it is really inspiring and reassuring! It would be great if more people in academia could be as honest and open like you :) thanks for all the videos :D
I'm really glad to hear that! I hope being honest about it makes it more obvious that most academics aren't breezing through- it's pretty tough for everyone! Still worth persisting :)
6:43 This is really important: published maths looks rigorous and linear, actual maths is detours, dead ends, mistakes, misconceptions. This really should be acknowledged and embraced. Suggestion: Publish a proof, but also dig out the crunched up pieces of paper from the bin and, as an appendix, publish those which are relevant too.
What a wonderful idea!
This is an excellent video, and *extremely valuable* to any prospective students interested in pursuing Maths(/Theo. Physics) research. These are often obscure fields difficult for students to imagine before jumping in! Very well put together video!
Beautifully inspiring, you have a gift.
You speak to many truths regarding research in the humanities, too. Thank you for sharing your experiences.
Thank you for sharing these experiences!
I'm really nervous about research at university. I'll be enrolling as a physics and math major this year and I'm pretty much terrified. This video is super helpful ngl
You're going to love it! There's ups and downs but overall it's just so fun to work on something so stimulating.
Good luck 😊 🙌
I've heard the same thing a lot that going on walks helps with creativity and figuring out problems that you were stuck on. I think it just helps boost my mood and give my head a break from the 'rigor' while exercising my creative side. I like to think about the possibilities and what I want to be true.
Great video so far and I find it really interesting because I'm looking to go into some combination of theoretical/mathematical physics and experimental work.
I'm a statistical geneticist and I related with this video so much. I also force myself to walk to and from work so i can cook up cool ideas. And the thing about dead-ends rings so true. So many times I thought I was onto something only to find out that it wasn't true in the end. But I learned a lot during the process! Thanks for making these videos :)
So grateful I can hear personal experience of a mathematician and a very good presentation of the process form high level perspective.
Thank you so much for this video! It was so interesting to hear your story.
I recently spent so long on a proof I had no idea how I had come up with the basic ideas for the proof in the first place. At then end of the day it was the result of a bunch of wandering, that had been catalogued along the way, and then resulting in something worthwhile. But the details of the wandering have left me.
That was a really great story, I am going into phd research as well so this was very helpful. Thanks!
I really appreciate your candor about missteps and failings. One of my biggest fears entering graduate school is making mistakes and feeling dumb amongst peers who are no doubt far more brilliant. You've taken the edge off that fear. I can also empathize with your creative process. For me it's riding public transport...bus or subway. Something about it liberates my creativity as I ponder physics. And that epiphany of an idea is so....satisfying. Then I rush to the literature to see if anything has been published on this idea. Inevitably there will be some research papers, but that's also satisfying as it validates my line of reasoning. It also gets one closer to the leading edge of knowledge.
Anyway, thanks for your courage in talking about your tribulations. We should all be more comfortable with taking chances and failing because that is where the learning happens.
Yeah, when you go in you only see other people’s successes. Everyone seems to be publishing so much! But soon you realise that failure is very common, and that’s fine. Completely normal (and very high probability) for a project to fail. Don’t worry about that at all :)
I know what you mean about being excited to see a thing you came up with in the literature already. You think, ah, so I’m thinking along the right lines!
This was one of the best videos you’ve made really conveying the feeling of what it’s like pursuing a PhD. What the thought process is like. Many of us dream to work at this capacity.
Everything you said hits so close to home. Maths or physics, incredible way to describe the process ;w;
You're channel is amazing! you're truly underrated
Awesome video. I'm studying informatics and programming now, and both in studying languages and the functioning of computer systems, which is all ultimately formal logic, though more on the 'engineering' side of things, I can definitely relate to discovery being a meandering creative path tempered by rigour. I also find some of my best ideas come about during my 'down time', when I'm just relaxing or doing something else, and they start out as hunches or intuitions about things that lead me in the right direction as I set out to prove them. Also, sometimes while in the process of correcting or evaluating someone else I realize my own flaws and mistakes, and figure out how to get things working. :)
this makes me so excited to pursue a research career in maths because i relate to the process so much and have my own walks of creativity,-----though admittedly only for high school mathematics.
Love your stories!
Here from Grant Sanderson's TED Talk and absolutely love this channel!! Kudos!!
Thank you for sharing your experiences! :D
THANK YOU. Every time I search for something maths academia related, I find something about being a maths teacher or undergrad maths. This video is exactly what I wanted.
I know it would be a lot of work, but I would really enjoy a video taking a journey from deciding to do maths for undergrad to doing maths research. I would personally watch the video at least 100 times.
As someone who has vacillated between the arts and sciences, I find something like this really fascinating: you describe maths and sciences and your thought processes around them in a way that will resonate with any artist. I’ve belatedly come to realise the two can be very complimentary, but in high school I felt like everyone was demanding I choose between them. Ultimately, because I didn’t feel maths and science would nurture my creativity, I chose the arts, but I wished I’d seen videos like yours way back then.
Great video. Thanks for sharing. I’ve been struggling with a result lately and this was inspiring.
Good luck!
I subscribed the moment I heard you say "the only way that you can justify feelings like this, is by noting that it would be ugly if that thing wasn't true". It is so true (even though it lead me astray before).
Great story. Story telling is a wonderful gift you have. Thank you.
This is excellent. Thank you for sharing.
Thank you so much for sharing your story 🙂
I've always loved maths, but I never was good enough to do it in uni unfortunately. Your story sounds amazing, it sounds like a really interesting experience.
I once took a set theory class and the final was to study the final part of Godel's incompleteness theorem in full rigor, and re-derive the concluding statements on an exam given a month to prepare, along with a couple smaller problems. What you said about "rediscovering" the author's path is exactly what that experience was like (although maybe on a smaller scale lol). There were times where I would obsess over the "hidden meaning" behind why Godel used this notation over that, or why he wrote something which seemed redundant, and sometimes it was a hint he left behind and sometimes it was just in my head, but to truly take what he left us and try to rediscover the thoughts he had was an amazing experience. Despite how stressful that month was, it was one of those times that, at the end of it all, only solidified my love for math and that this is what I want to do with my life. Great video :)
An excellent video! I was strongly reminded of Dr. Albert Einstein's struggle with the mathematics of GTR and the ideas of Hermann Weyl that he thought led to an unphysical second clock effect without the support of the Geometric Calculus of multivectors and rotors.
I'm really glad you stress the importance of taking a walk and just thinking about things. I was once trying to make a mathematical proof, and I found that while laying in bed trying to sleep or just waking up was when I had a clear enough mind to make a ton of progress on it, even though when I would scurry to write it all down, it would invariably be wrong. I never felt bad about those thoughts being wrong, though, because I had nothing to read off of, no calculator to easily do math, and no guarantee that I was even solving the correct problem! I had once found that I made a workable proof, only to find that I proved something entirely different! Still, these times of thought were incredibly valuable for thinking about the problem in different ways and increasing my understanding of it, which made working with the proof a lot easier and led the way to different ideas I could potentially use. Just as important as learning the fundamentals and standard equations is letting your brain have freedom to be different!
As Erdos would say, I'm dead - but I haven't left yet. Thanks for reminding me of all those idle moments consumed by intuition and truth.
Great video, and very instructive story! And humbling, too.
I watched your videos since I had to finish my first project about Fourier transform in year 1. Now I am going graduate from physics MPhil. Time flies, and I hope you have a great time on long journey of physics research.
Wow! That's amazing, I hope you're enjoying yourself!
I love your stories. Really cool. As a designer I try to approach a new idea with 3 completely different approaches, in the early stage of ideation. When I was at high school I spent a year developing mechanical creation skills by drawing everything mechanical I knew on butchers paper ( a wide roll of cheap white paper) endlessly pulling out a new sheet to start with a new sketch. That was my equivalent to your walks. For me though showering in the dark is the best creative space. There is something about warm flowing water that frees the mind, its a happy space.
I really can’t see how to be creative with maths,….I really need to,…but its not a happening thing for me, leaving me in awe as to how others such as yourself can be. Serendipity is a powerful friend. One of my best (and simplest) inventions came about through scraps of knowledge gained over years, and work situations that allowed me to collect some exotic materials, so that when a problem arose needing a solution, all of the bits were there and it came together in minutes. And innovations with the exotic properties of that material, carbon fibre, keep coming.
Following your journey is fun. Keep smiling.
12:02, reinterpreting two other results and seeing the connection between them is creativity (and it is work).
I couldn't reach my "mirage" during my undergrad research thesis. It was scary - and very stressful - but also somehow got me excited over the small results I managed to derive. This video inspires me to overcome the fear and embrace the excitment in the journey.
Excellent summary of the process. Matches completely with my own experience and things that I have said (a few in writing). Maybe you have some thoughts on how to select worthwhile problems and how to know when to give up on them, because that is something I have been very bad at over the course of my career. The people who are able to achieve the most overall success during a lifetime of mathematical research are, in my view, those who are able to select the most important problems that are within their capability to solve. Not an easy feat!
You word it very eloquently. I always have a hard time describing why math is art.
Really appreciate it if you could share more on your thought processes on doing math research!
*Hey lovely to see you back*
We love your content
Thank you!
Mathematics is an art like music or painting. It is the art of reasoning which is unfortunately not recognized by our society
This video is so honest and accurate, bravo!
Good to see other math students going through the same difficulties as me. I was researching too in quantum computing and quantum algoritms as an MSc math student this year and it was a really similar experiencr that you describe. I had discovered a new algorithm, that was a little better in some aspects, but only worked for a small number of groups. I worked months and months trying to generalize it, but in the end i couldn't and I got pretty convinced that it's probably impossible to. Luckily in the end I could publish the small results I got. Resarching is really frustrating sometimes, but the feeling that you get when solving a new problem noene has solved yet is something that's worth all that and even the progress of coming up with the proofs piece by piece is great fun too.
By the way, if you read this, I would be really intrested in the paper you published in the end.
I'm glad to hear you were able to publish something from that. It's so frustrating when this sort of thing happens. What algo where you working on :)? What are you doing now?
The result was an extension of the Gottesman Knill theorem: arxiv.org/abs/1806.03200
@@LookingGlassUniverse It was a new algorithm for a special case of the hidden shift/hidden subgroup problem, that was polynomial in a sense and used little space. I fell in love with quantum computing and quantum complexity theory around a year ago, but since i got stuck i decided to move to game theory for this semester. It's also really beatiful and quite cool. And there is field called quantum game theory, so I don't have to go that far from the quantum world.
Also thanks again for your video, it really helped to boost up my confidence in researching and that that's what I want to do in my life. Hope you will be succesful too in the things you work on! :) I will read the paper you sent, it looks interesting.
Have a nice day!
Wow! Excellent explanation of what math really is! More people need to see this!
You have a very interesting view on the world. I love your videos.
This is so excellent! This way of talking about math should have a big impact on young minds who would otherwise associate it with a joyless stern old guy at the front of a class. It sounds difficult in the way that important things are difficult, but inspiring at the same time.
wow! amazing....I am an aspiring theoretical physicist and I have got a lot confidence from this content..thanks!
This is an interesting, original presentation. I enjoyed watching it. One thing I learned from you is that mathematicians need a "muse" just like artists do. I will define muse here as an activity, experience, mental and/or physical state, a narrative or some body of knowledge that enables us to be significantly more creative and productive. You are lucky Mithuna. Those morning walks of your's are a good muse. They work for you. Moreover, these walks are clearly not super rigorous and do not endanger you in any way. With some other people it is different. They have a muse that is actually quite dangerous. Some of the world's greatest scientists, engineers and artists actually did their most important work when they were hovering on the "knife edge" between life and death. The great British naturalist Alfred Russell Wallace is one example. His greatest discovery, the Wallace Line, came (in Indonesia) when he was critically ill with malaria. Another example is James Eads, one of America's greatest engineers. Eads' most important breakthroughs came when he had pushed himself so hard that he was literally on the brink of collapse from exhaustion. Indeed, on two occasions, after coming up with brilliant solutions to formidable engineering problems, Eads did collapse from exhaustion and only recovered after months of rest. Yet another example is the famous, Pulitzer Prize winning Japanese photographer, Kyoichi Sawada. He took his greatest photos during ferocious, extremely dangerous firefights during the Vietnam War. Kyoichi literally needed this proximity to violence and death to do his best work. Unsurprisingly, Kyoichi's muse eventually killed him. He was shot and killed (in 1970) by renegade communist soldiers while working in Cambodia.
As a physics student myself, I agree... excellent video!!
Tangent: I've haven't heard that many people categorize engineering as a creative profession. What? Don't engineers design and build stuff? That's not creation? And what about problem-solving (what I consider the core of engineering)-it's almost completely creativity.
Here we see that in math. Way to go dude!
Yes!! "math is a rollercoaster" as you said with many emotions while thinking about it, from தமிழ்நாடு, huge fan..
நன்றி!
Been looking forward to this.
Took a little longer than I hoped :P
this is beautiful to think about
I absolutely relate to having more interesting ideas when I can't write equations. I don't really do many equations these days because I left academia and became a coder, but the same idea absolutely applies there, the big picture stuff can sometimes be figured out without writing any code and maybe you've got a great algorithm in a fraction of the time, or you've managed to rule out a bad idea faster than you might. It, yes, you imagine a great idea that doesn't work. But you can't win 'em all.
Very insightful, thanks!
Id like it. The eureka is like a deja vu or reminiscense. It's result of the deep functions of the Brain. The rational thoughts comes from the "irrational" side of the Brain. There are many amazing contributions from neuroscience about that. Thanks for your testimony
Another informative video! That mix of creativity and rigor sounds scary for sure lol. Could you do one about the distinctions between applied and pure math from your perspective? There's such a spectrum of what's considered applied math. I'm into "theoretical"/mathematical biology and I barely see any proofs, but on the other end there are papers on PDE, topological data analysis, dynamical systems, etc. that are so dense (not that I can actually read them).
My little brother seems to be pretty stuck on pure math because of the rigor, but I think it'll be good for students like him to see there are parts of applied math research that are just as "challenging", if not just for their own careers' sake haha.
Such a great question! I feel unqualified for it though, since applied math is so vast a field, and I only know my little corner :'( But I agree that more people should know about this topic! Many a future video..
@@LookingGlassUniverse I really hope you do! We all probably give you a lot more credit than you do yourself lol. You said in one of your vids you loved learning abstract algebra. I'm not sure how you see your channel's trajectory, but do you ever think you'll do vids on topics from math foundations (abstract algebra, functional analysis, topology, ...) that you think are interesting/useful?
@@mathieumaticien Yeah, that'd be so fun!! Thanks for the great ideas :)
Got somehow similar experience while doing PhD in theoretical physics. In my case I was searching for exact equations and solutions governing microstructure of sedimenting suspensions and I got a tons of false routes, tons of failures ending in proving nothing but in the end I've found a route to new hierarchy of equations for microstructure. But in my case it was not the hierarchy for long range as I wanted at the beginning but for short range ones. Great unique results but still not the one I wanted the most :D Anyway I FULLY AGREE that theoretical, mathematical work is super exiting and kinda unpredictable :) True rollercoster :)
It's the purest ecstasy to see you back!
Aw! Thank you!
I have my undergrad degree in Economics (focus on Econometrics). We used a lot of applied math and statistics. A lot of it was trial and error just trying to find something to study and create a hypothesis. It's just as much art as it is rigorously following known formulas and theories on the macro level.
its a shame I finally began to really appreciate maths towards the latter of my Computer Engineering Degree, Why is discrete math not thought in a highschool -middle school curriculum. This was the class that helped set the foundation that my math curriculum has always lacked. Being introduced to logical proofs and axioms made me appreciate the complexity and pursuit for true facts. Its lovely! and I always enjoy talking to math majors around my uni
I do agree with what your findings, But I also agree, that if you make a discovery in maths, that the mathematician is to use the rules of simplification. This is a must. The simplification is the simplest way of explaining your discovery and you won't get lost in it because your using the basics and principles of the related symbol for your example. If you have an idea that you would like to reach, the principle of mathematics is that, you would meet at the starting point which would be the idea (The Idea). Then you would simplify the journey to and from the idea you discovered. You have to be able to reach the idea and explain it in its most simplistic form by its means. If you remember that, you will never get lost, and your discovery will only be redefined simpler, but the basic principle of your idea, will still stand always.
Thank you for sharing....
Thank you for sharing wonderful experience...;
Nice (and important) distinction around 9:00: realwork vs administration.
You are a great storyteller.
really cool videos!! keep making more please!!
Thanks for your insight.
Amazing video. I so wish I could have seen this when I was in grad school!
Rigorousness versus creativity, you could not say that better. I'm now a postdoctoral researcher in "pure mathematics" (homological algebra, category theory, noncommutative algebraic geometry). Looking back to my PhD years, I think that my main error was to be over-reliant on trying to understand everything in the most rigorous possible way, sacrificing intuition. I practically had to learn to trust my raw ideas, to allow myself to devise "vague arguments" before rigorous ones. I hope it was not too late to "save" my career. ;)